Store Separation Autopilot

ABSTRACT

A method and apparatus are presented for guiding a store, represented by a dynamic system having transitory nonlinear characteristics, between release from a platform and an activation of a mission autopilot along an optimal path. A nominal reference trajectory is determined that optimizes a desired performance index for the dynamic system using optimal control theory. A feedback control system is implemented that optimizes an original performance index to second order in a presence of disturbances along the optimal path using neighboring optimal control. The feedback control system converges to a linear time invariant regulator approaching the desired operating condition along the optimal path. Finally, control of the store is transitioned to the mission autopilot.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. application Ser. No.14/332,529, entitled “Store Separation Autopilot,” filed on Jul. 16,2014, which claims the benefit of and priority to U.S. ProvisionalApplication Ser. No. 61/856,780, entitled “Store Separation Autopilot,”filed on Jul. 22, 2013, the entireties of which are incorporated byreference herein.

RIGHTS OF THE GOVERNMENT

The invention described herein may be manufactured and used by or forthe Government of the United States for all governmental purposeswithout the payment of any royalty.

BACKGROUND OF THE INVENTION

Field of the Invention

The present invention generally relates to guidance and control systemsand, more particularly, to guidance and control systems for anexpendable store.

Description of the Related Art

Tactical fighter and bomber aircraft have been used to carry and deliverordinance since shortly after the dawn of aviation. In the earlieststages of air combat, separation of stores from the parent aircraft wasof little concern. However, during the Vietnam War, the employment ofheavy stores from larger jet-powered aircraft began to presentdifficulties for aircraft-store compatibility. Specifically, scenariosin high-speed flight were encountered where the released store failed toseparate cleanly from the aircraft and instead became a projectilethreatening the aircraft and on occasion re-contacting the aircraft inflight causing catastrophic damage and loss of life.

A store released from an aircraft in flight must traverse a nonuniformand unsteady flow field that may include complex shock interactions,large velocity gradients, regions of locally separated or reversedairflow, and severe flow angularity in the form of sidewash anddownwash. Stores released from an internal weapons bay may also besubjected to a wake disturbance from the spoiler, dynamic pressure andvelocity gradients across the shear layer, high frequency vibrations dueto acoustic noise, and large perturbations in flow properties due tocavity oscillations. Although the region of nonuniform flow near theaircraft is exceedingly small compared to the full length of the storeballistic or fly-out trajectory, the effects are significant.

Store separation engineering, a subset of aircraft-store compatibility,is concerned with the flight characteristics of a store in proximity ofthe aircraft and other stores. Ground test, flight test, simulation, andanalysis procedures have been developed which largely address thesafety-of-flight issues first encountered in the Vietnam era. In mostcases, the store can be ejected away from the aircraft with a sufficientvertical velocity and nose-down pitch rate to ensure safe separation.However, with the advent of smart weapons, standoff capabilities, andfocused lethality the challenge in successful store separation hasshifted from safety to acceptability. Whereas an unsafe separation maythreaten the parent aircraft, an unacceptable separation may result in afailed mission or significant collateral damage due to guidanceproblems, loss of control, or damage to the store caused by theseparation transients.

Modern sophisticated “smart” weapons are equipped with sensitive onboardelectronics including inertial measurement systems, GPS units, sensors,seekers, and guidance computers. Standoff capability (the desirableability to release a munition far away from the intended target) hasresulted in complex aerodynamic shapes with neutral dynamic stabilitymargins designed for maximum glide performance and minimal energy loss.Focused lethality (the desirable ability to destroy a designated targetwhile minimizing collateral damage) has resulted in munitions that aresmaller and lighter and therefore more dramatically affected by theexigent flow field surrounding the aircraft in flight. These tendencieshave increased the sensitivity to separation-induced transients,potentially leading to large angular rates and attitudes, excessiveenergy loss, sensor saturation, structural limits, or departure fromstable flight modes. A challenge in store separation is thus to ensuresafety while also maintaining acceptability across the flight envelope.

Modern munitions are designed with an onboard guidance and controlsystem to enable precise engagement of the intended targets. However,the control system is not usually activated until the store issufficiently far away from the aircraft to avoid any potentialinterference. Often, the separation-induced transients result in largeperturbations from the desired flight attitudes that require a dedicated“rate-capture” phase for recovery before the munition can begin thefly-out trajectory. In the relatively few cases where the autopilot isengaged earlier (to prevent build-up of irrecoverable rates andattitudes), the mutual aerodynamic interference between the store andaircraft is neglected in the autopilot design leading to increased riskthrough reduced confidence in simulation capabilities and potentiallyunsafe behavior of the autopilot reacting to flow field perturbationswithout consideration of the nearby aircraft.

Accordingly, there is a need in the art for a transitional controlsystem that accounts for separation-induced transients to guide a storealong a preferred trajectory.

SUMMARY OF THE INVENTION

A method of guiding a store between release from a platform and anactivation of a mission autopilot along an optimal path is presentedwhere the store is represented by a dynamic system having transitorynonlinear characteristics. A nominal reference trajectory is determinedthat optimizes a desired performance index for the dynamic system usingoptimal control theory. A feedback control system is implemented thatoptimizes an original performance index to second order in a presence ofdisturbances along the optimal path using neighboring optimal control.The feedback control system converges to a linear time invariantregulator approaching the desired operating condition along the optimalpath. Finally, control of the store is transitioned to the missionautopilot.

In an embodiment incorporated into an air-to-ground guided munition, aflight management system is configured to manipulate flight controlsurfaces of the air-to-ground munition. The flight management systemincludes a store separation autopilot and a mission autopilot. The storeseparation autopilot is activated when the air-to-ground munition isreleased from a platform. Furthermore, the store separation autopilot isconfigured to determine a nominal reference trajectory that optimizes adesired performance index for the dynamic system using optimal controltheory. Similarly, a feedback control system is implemented in the storeseparation autopilot that optimizes an original performance index tosecond order in a presence of disturbances along the optimal path usingneighboring optimal control and corresponding manipulation of the flightcontrol surfaces. The feedback control system converges to a linear timeinvariant regulator approaching the desired operating condition alongthe optimal path. Then, control of the flight management system of theair-to-ground munition is transitioned to the mission autopilot.

Additional objects, advantages, and novel features of the invention willbe set forth in part in the description which follows, and in part willbecome apparent to those skilled in the art upon examination of thefollowing or may be learned by practice of the invention. The objectsand advantages of the invention may be realized and attained by means ofthe instrumentalities and combinations particularly pointed out in theappended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate embodiments of the invention and,together with a general description of the invention given above, andthe detailed description given below, serve to explain the invention.

FIG. 1 is an exemplary illustration of an unsafe and unacceptablerelease of a store;

FIG. 2 is an exemplary illustration of a safe and unacceptable releaseof a store;

FIG. 3 is an exemplary illustration of a safe and acceptable release ofa store;

FIG. 4 is an exemplary diagram of a store illustrating a flightmanagement system and control surfaces;

FIG. 5 is a block diagram illustrating a feedback control loopconsistent with embodiments of the invention;

FIG. 6 is a block diagram illustrating components of a store separationautopilot consistent with embodiments of the invention;

FIGS. 7A-D are graphs of optimal flight trajectory attributes for anexemplary subsonic test mission;

FIGS. 8A-D are graphs of optimal flight trajectory attributes for anexemplary supersonic test mission;

FIGS. 9A-D are graphs of a comparison of optimal and flight testtrajectory attributes for the exemplary subsonic test mission of FIGS.7A-D; and

FIGS. 10A-D are graphs of a comparison of optimal and flight testtrajectory attributes for the exemplary supersonic test mission of FIGS.8A-D.

It should be understood that the appended drawings are not necessarilyto scale, presenting a somewhat simplified representation of variousfeatures illustrative of the basic principles of the invention. Thespecific design features of the sequence of operations as disclosedherein, including, for example, specific dimensions, orientations,locations, and shapes of various illustrated components, will bedetermined in part by the particular intended application and useenvironment. Certain features of the illustrated embodiments have beenenlarged or distorted relative to others to facilitate visualization andclear understanding. In particular, thin features may be thickened, forexample, for clarity or illustration.

DETAILED DESCRIPTION OF THE INVENTION

Combat aircraft utilize expendable stores such as missiles, bombs,flares, and external tanks to execute their missions. Safe andacceptable separation of these stores from a parent aircraft isessential for meeting mission objectives. In many cases, the employedmissile or bomb includes an onboard guidance and control system toenable precise engagement of a selected target such as another aircraftfor an air-to-air guided munition or a land based target for anair-to-ground guided munition. Due to potential interference, theguidance and control system is generally not activated until the storeis sufficiently far away from the aircraft. This delay may result inlarge perturbations from a desired flight attitude caused by separationtransients, significantly reducing effectiveness of the store andjeopardizing mission objectives.

The flow field characteristics may cause the store to exhibit behaviorthat compromises the safety of the airframe and crew or that compromisesthe effectiveness of the store itself. Prediction of the flightcharacteristics of the store in the vicinity of the aircraft istherefore vitally important for ensuring the safety and effectiveness ofthe release. Modeling and simulation capabilities also play an integralrole in the cost-effective assessment of separation characteristics fora range of aircraft and store configurations throughout the aircraftflight envelope.

Successful store separation is a balance between two competingobjectives. First, a successful store separation trajectory must be safeand not exhibit any threatening motion toward the aircraft, asillustrated in FIG. 1. In some cases, lateral motion is the primaryconcern due to tight tolerances between the store and adjacent aircraftcomponents or additional stores. However, in most cases, safe separationis dominated by the vertical translation of the store. If the storeescapes the aircraft flow field with a monotonically increasing verticalvelocity, then the trajectory is considered safe. If the store hesitatesor begins to flyback to the aircraft, the trajectory is consideredunsafe. Due to uncertainties in separation prediction methods andvariations in store and aircraft properties, flight-testing of unsafetrajectories is usually avoided altogether.

In most cases, the store is launched from an ejector providing aninitial vertical velocity. In order to flyback, the store must generateenough aerodynamic lift to first arrest the vertical velocity and thenbegin translation in an upward direction. Thus, flyback is alwayspreceded by a significant duration at a positive angle of attack. Formost stores, limiting the angle of attack can ensure a safe separation.The safety margin is increased when the angle of attack is negativethroughout much of the trajectory, generating aerodynamic forces in thedirection of translation and accelerating the store away from theaircraft.

A second criterion for a successful separation is that the trajectorymust be acceptable, i.e. the transitory effects of the separation mustnot compromise the ability of the store to achieve a specified mission.An unsafe trajectory cannot be acceptable, but a safe trajectory may beunacceptable, such as the trajectory illustrated in FIG. 2. Therefore,safety is a subset of acceptability. In comparison to safety, it isgenerally more difficult to quantify and ensure acceptability. However,acceptability can be adequately addressed by the following fourconditions.

-   -   Store total aerodynamic angle of attack should not exceed the        specified range for which the store autopilot has been designed        to function properly.    -   Angular rates and accelerations should not exceed the specified        range for which the onboard instrumentation is sufficient to        measure.    -   Control inputs should not exceed the specified capability of the        control actuators.    -   Total aerodynamic loads should not exceed the safety margins for        the structural integrity of the store and empennage.

Precise statement of the acceptability conditions requires considerationof a specific system. In general terms, acceptability can be achieved bykeeping the total angle of attack and angular rates low and by limitingcontrol effort. A narrow but useful condition for acceptability,especially in control system design, is to require the state and inputbe maintained within a certain predefined operating range over which thecontrol system has been designed to function properly.

Finally, it is recognized that a separation autopilot is a transitionalcontrol system, intended to guide the store through the nonuniform flowfield and transfer the control to the mission autopilot. As such, anobjective of a separation autopilot is to safely drive the store to anear-equilibrium state at or before the transition to the missionautopilot such as illustrated in FIG. 3. Therefore, it is desirable notonly for certain components of the state to be near zero, but also forcertain components of the derivative of the state to be near zero.

Previous studies have highlighted the use of active control to improveseparation characteristics; however, embodiments of the invention arethe first to consider guidance and control specifically for storeseparation. Guidance herein refers to the determination of a preferredpath from release to a stable trimmed flight condition with explicitdependence on aerodynamic interaction between a store and an aircraft.Control herein refers to a manipulation of aerodynamic forces usingdeflections of control surfaces 12, 14, such as those illustrated inFIG. 4, to steer the store 10 along the preferred trajectory in thepresence of disturbances.

Flight vehicles, such as aircraft and guided stores 10, use flightmanagement systems (FMS) 16 to achieve guidance and control throughoutthe flight profile. The pilot or FMS will frequently switch betweenautopilots that perform different functions, such as altitude hold,climb/descent, bank-to-turn, etc. In this context, a store separationautopilot 18 is a transitional control system, designed to effectivelytransfer the store from release to a stable trimmed flight condition.This transitional duration may be approximately one second after releasein some embodiments. Spatially variant aerodynamic characteristics areaccounted for through the nominal optimal trajectory 22 as seen in anexemplary feedback control loop 20 in FIG. 5. Response to varyinginitial conditions and flow field disturbances are accounted for usingfull-state feedback based on neighboring optimal control techniques 24.

A design of an exemplary store separation autopilot 30, as illustratedin FIG. 6 and consistent with embodiments of the invention, begins witha technical description of an aircraft and a store 32, includingcharacterization of aerodynamic interference effects usually acquiredfrom wind tunnel testing or numerically from computational fluiddynamics. Additionally, the autopilot 30 may be designed to include aspecific set of safety and acceptability criteria 34. The dynamic system32 and safety and acceptability criteria 34 form the inputs to the storeseparation autopilot 30 formulation.

Using the aircraft/store input data 32, a high-fidelity aircraft/storenumerical model 36 may be constructed using conventional techniques forstore separation trajectory prediction. A store aerodynamic model 38 isa subset of the high-fidelity aircraft/store model 36. Linearization ofthe store aerodynamic model at a particular trim condition leads to alocal far-field model 40 that neglects aerodynamic interference from theaircraft. The linear far-field model is used to determine the constantLinear Quadratic Regulator (LQR) feedback control gains (K_(LQR)) 42,using conventional LQR design techniques for a linear time-varyingsystem. The end-point Mayer cost S_(f) may be readily determined from asolution of the algebraic Riccati equation. The safety/acceptabilitycriteria 34, which may be formulated as Q, R matrices, and the end-pointMayer cost S_(f) describe the optimization cost functional 44,J(S_(f),Q,R). This formulation is unique to an Infinite HorizonNeighboring Optimal Control approach.

An optimal trajectory 46 may be found computationally using the highfidelity model 36 by solving a Hamiltonian Boundary Value Problem. Theresulting nominal trajectory provides an optimal flight path andopen-loop control inputs. Given this optimal trajectory 46, the highfidelity model may be linearized along a prescribed flight path,resulting in a local model 48. The local model 48 may then be used tocompute time-varying feedback gains 50 by solving the differentialRiccati equation. Given the optimal trajectory 46 and feedback controlgains 50, the store separation autopilot 52 may be implemented using astandard feedback control loop, such as that illustrated in FIG. 5above, with the dynamic system 26 incorporating the appropriate modelsdiscussed in relation to FIG. 6.

Application of neighboring optimal control to store separation isstraight forward. A quadratic cost functional, given by Equation (1) issufficient for this investigation, where Q is a constant positivesemi-definite matrix Q≧0 and R is a constant positive definite matrixR>0. The weighting matrices Q and R are chosen by a user to influence amagnitude of a state and control vector, respectively. Matrix S_(f)≧0 isspecified by the user to achieve satisfactory terminal conditions.

$\begin{matrix}{J = {{\frac{1}{2}{x\left( t_{f} \right)}^{T}S_{f}{x\left( t_{f} \right)}} + {\frac{1}{2}{\int_{t_{0}}^{t_{f}}{\left( {{x^{T}{Qx}} + {u^{T}{Ru}}} \right){dt}}}}}} & (1)\end{matrix}$

Using the quadratic cost functional, first order optimality conditionswithout terminal constraints are stated in Equations (2) through (4).

{dot over (x)}(t)=f(x(t), u(t)), x(t _(o)) specified   (2)

{dot over (λ)}(t)=Qx(t)−f _(x) ^(T)(t)λ(t), λ(t _(f))=S _(f) x(t _(f))  (3)

u(t)=−R ⁻¹ f _(u) ^(T)(t)λ(t)   (4)

Due to the difficulty of modeling store separation aerodynamics, it isdesirable to isolate the aerodynamic terms appearing in the stateequation {dot over (x)}(t)=f (x, u). This allows optimality equations tobe used with a variety of aerodynamic models. Recognizing that theaerodynamic terms are also functions of the state and control, the stateequations can be written in functional form as shown in Equation (5),where C_(F)(x, u) and C_(M)(x, u) are the aerodynamic force and momentcoefficients, respectively.

{dot over (x)}(t)=f(x, C _(F)(x,u), C _(M)(x,u))   (5)

Using the notation in Equation (5), the Jacobian matrices in Equations(3) and (4) can be expanded as follows.

$\begin{matrix}{f_{x}\overset{\Delta}{=}{\frac{\partial{f\left( {x,C_{F},C_{M}} \right)}}{\partial x} = {\frac{\partial f_{0}}{\partial x} + {\frac{\partial f}{\partial C_{F}}\frac{\partial C_{F}}{\partial x}} + {\frac{\partial f}{\partial C_{M}}\frac{\partial C_{M}}{\partial x}}}}} & (6) \\{f_{u}\overset{\Delta}{=}{\frac{\partial{f\left( {x,C_{F},C_{M}} \right)}}{\partial u} = {\frac{\partial f_{0}}{\partial u} + {\frac{\partial f}{\partial C_{F}}\frac{\partial C_{F}}{\partial u}} + {\frac{\partial f}{\partial C_{M}}\frac{\partial C_{M}}{\partial u}}}}} & (7)\end{matrix}$

Equations (6) and (7) can be written more concisely using subscriptnotation to represent partial differentiation, where the notation f_(x)_(o) and f_(u) _(o) implies the derivative is taken while holding theaerodynamic coefficients constant.

f _(x) =f _(x) _(o) +f _(C) _(F) C _(F) _(x) +f _(C) _(M) C _(M) _(x)  (8)

f _(u) =f _(u) _(o) +f _(C) _(F) C _(F) _(u) +f _(C) _(M) C _(M) _(u)  (9)

The matrices C_(F) _(x) and C_(M) _(x) represent the aerodynamicstability derivatives, and the matrices C_(F) _(u) and C_(M) _(u)represent the aerodynamic control derivatives. These matrices may bedetermined analytically when a parametric form of the aerodynamic modelis available, often as a result of system identification. Alternatively,they can be estimated numerically using finite differencing or analternative numerical recipe.

Beginning with the necessary conditions in Equations (2) through (4),the linear differential equations for a neighboring extremal aresummarized in Equations (10) through (12).

δ{dot over (x)}(t)=f _(x) δx+f _(u) δu   (10)

δu(t)=−R ⁻¹ f _(u) ^(T) Sδx   (11)

{dot over (S)}(t)=−Sf _(x) −f _(x) ^(T) S+Sf _(u) R ⁻¹ f _(u) ^(T) S−Q,S(t _(f))=S _(f)   (12)

Equations (10) through (12) are a compact set of differential equationsthat can be used to implement a Store Separation Autopilot thatminimizes the original cost function to second order in the presence ofdisturbances along a predetermined optimal trajectory. The matrixRiccati equation (12) is evaluated along an optimal trajectory todetermine feedback gains K(t)=−R⁻¹f_(u) ^(T)S and the results are storedalong with the nominal state and control, x*(t) and u*(t). Theneighboring optimal control input can be determined real-time using feedforward of the nominal control plus feedback proportional to thedeviation of the measured state from the reference trajectory,u(t)=u*(t)−K(t)δx(t).

Aerodynamic characteristics of a store in the vicinity of the aircraftare inherently nonlinear. Aerodynamic nonlinearities appear throughlarge flow field gradients near the aircraft as well as decay of theaircraft effects in far field conditions. Thus, the store transitionsthrough a time (or spatially) variant nonlinear regime and rapidlyapproaches a trimmed freestream flight condition that can be adequatelyapproximated by time invariant linear behavior. One approach tocontrolling a store in these two disparate flight regimes is to switchbetween a nonlinear time variant controller and a linear time invariantcontroller. Another approach is to design a single control system thataccounts for the nonlinear flight regime and converges to a linear timeinvariant controller in far field conditions. The latter approach isadopted here in a process herein referred to as Infinite HorizonNeighboring Optimal Control, which is graphically illustrated aselements 36-50 in FIG. 6.

The neighboring optimal feedback gains K(t)=−R⁻¹f_(u) ^(T)S may bedetermined in part by the solution to the matrix differential Riccatiequation (12). The Jacobian matrices f_(x) and f_(u) are in generaltime-varying. For store separation, these matrices result fromlinearization along a predetermined trajectory and vary with time and/ordistance from the aircraft due to the nonlinear aerodynamiccharacteristics. However, as the distance between the store and aircraftbecomes large, the effect of the aircraft flow field becomes negligibleand the Jacobian matrices converge to constant freestream quantities,denoted here as F and G.

$\begin{matrix}\left. {\lim\limits_{t\rightarrow t_{f}}{f_{x}(t)}}\rightarrow F \right. & (14) \\\left. {\lim\limits_{t\rightarrow t_{f}}{f_{u}(t)}}\rightarrow G \right. & (15)\end{matrix}$

In this limiting case, the matrix differential Riccati equation (DRE)approaches a constant solution, resulting in an algebraic Riccatiequation (ARE) which may be solved numerically to yield S_(f).

0=−S _(f) F−F ^(T) S _(f) +S _(f) GR ⁻¹ G ^(T) S _(f) −Q   (16)

The solution to the ARE can be used to determine the constant feedbackgains K_(f)=−R⁻¹G^(T)S_(f). The resulting linear time invariant controlsystem is mathematically equivalent to a Linear Quadratic Regulator(LQR).

Returning to the original quadratic cost functional in Equation (1), thematrix S_(f) is used to denote a user-specified weighting matrix thatdetermines an end point (Mayer) cost. Choosing the Mayer cost to beconsistent with the solution to the ARE results in a time varying gainmatrix that approaches a constant quantity as the system converges to atime invariant system. The time invariant gains can be used to maintainthe system near the desired operating condition indefinitely.

$\begin{matrix}\left. {\lim\limits_{t\rightarrow t_{f}}{K(t)}}\rightarrow K_{f} \right. & (17)\end{matrix}$

Thus, Infinite Horizon Neighboring Optimal Control (IHNOC) consists ofthree sequential steps as illustrated in the block diagram 20 in FIG. 5.First, optimal control theory is used to determine a nominal referencetrajectory that optimizes a desired performance index for a dynamicsystem with transitory nonlinear characteristics in block 22. Next,neighboring optimal control is used to implement a feedback controlsystem that optimizes the original performance index to second order inthe presence of disturbances along the optimal path in block 24.Finally, as the system approaches an operating condition that isadequately represented by a linear system model in block 26, thefeedback controller converges to a linear time invariant regulator thatmay be used to keep the system near the desired operating conditionindefinitely. While this may be possible, embodiments of the storeseparation autopilot are envisioned to be a transitional control systemthat is active for a relatively short period of time between the releaseof the store and a transition to the flight management system thatdirects the store to its target destination.

In a practical application of the store separation autopilot, an exampleis presented below of a representative store separating from anexemplary F-16 aircraft, such as those graphically illustrated in FIGS.1-3. The F-16 is a multi-role supersonic fighter aircraft originallydeveloped by General Dynamics. The F-16 may be configured to anair-to-air or air-to-ground configuration and may be equipped to carry arange of external stores. Separation of a representative store from theF-16 is illustrated in FIG. 3. Wind tunnel testing was conducted toquantify freestream aerodynamic characteristics of a store, as well asspatially variant effects of the aircraft flow field. Flight testing wasconducting using an uncontrolled inert separation test vehicle todemonstrate save and acceptable separation characteristics. For thepresent example, wind tunnel data were used to determine a nonlinearspatially variant aerodynamic model for trajectory optimization. Theoptimization was accomplished using full nonlinear six degree of freedomequations of motion and simulations were conducted using conventionalstore separation trajectory prediction methods.

FIGS. 7A-D and 8A-D illustrate optimal trajectories computed usinginitial conditions, mass properties, and flight conditions from twoflight test missions, a subsonic mission—Mach 0.9/550 KCAS/4800 ft, anda supersonic mission—Mach 1.2/600 KCAS/18,000 ft, respectively. Theperformance metric was specified using a quadratic cost functional withweighting factors selected to minimize the store incidence anglesα_(s)(t) and β_(s)(t) and the roll rate p(t) without excessive controleffort. The pitch and yaw rates were also reduced through kinematiccorrelation with the derivatives α_(s)(t) and β_(s)(t). The specificvalues of the Euler angles ψ(t), θ(t), and φ(t) were not of particularconcern, only the incidence angles and roll are were included in thecost function.

In comparison to the subsonic trajectory, the supersonic flow fieldresulted in a larger nose-down aerodynamic pitching moment nearcarriage. The optimal control used a maximum control authority of −10degrees to arrest the pitch rate and angle of attack. The strongerflowfield resulted in higher deviations in pitch rate throughout thetrajectory. Even in these adverse conditions, the optimal controlsuccessfully brought the store to a stable trimmed flight conditionwithin a 1 second time interval. These results indicate that the controleffectively transfers the store from aircraft carriage to stable trimmedflight in an optimal manner.

FIGS. 9A-D show a comparison between an uncontrolled (jettison) flighttest trajectory for the subsonic test flight mission and the computedoptimal trajectory illustrated in FIGS. 7A-D. FIGS. 10A-D show a similarcomparison of the supersonic flight test with the optimal trajectoryfrom FIGS. 8A-D. In both cases, the controlled separation shows adramatic improvement over the flight test trajectory in terms ofacceptability margins. For the supersonic test mission, the maximumangle of attack was reduced from α_(max)=−20 deg to α_(max)=−5 deg andthe maximum pitch rate was reduced from q_(max)=130 deg/sec toq_(max)=28 deg/sec.

Comparison of the optimal trajectories between flight conditions is alsovaluable. Whereas the flight test trajectories are dramaticallydifferent between the subsonic and supersonic flight conditions, theoptimal trajectories are very similar. The optimal control program notonly provides a measureable improvement in safety and acceptability, butit also assists in reducing the variability in trajectorycharacteristics between flight conditions. The uniformity between flightconditions is an advantage for ensuring safe and acceptable employmentacross the flight envelope.

While the present invention has been illustrated by a description of oneor more embodiments thereof and while these embodiments have beendescribed in considerable detail, they are not intended to restrict orin any way limit the scope of the appended claims to such detail.Additional advantages and modifications will readily appear to thoseskilled in the art. The invention in its broader aspects is thereforenot limited to the specific details, representative apparatus andmethod, and illustrative examples shown and described. Accordingly,departures may be made from such details without departing from thescope of the general inventive concept.

What is claimed is:
 1. A method of guiding an air-to-ground guidedmunition, between release from a platform and an activation of a missionautopilot along an optimal path, the method comprising: determining anominal reference trajectory that optimizes a desired performance indexfor the air-to-ground guided munition using optimal control theory;implementing a feedback control system that optimizes an originalperformance index to second order in a presence of disturbances along anoptimal path using neighboring optimal control; converging, via thefeedback control system, to a linear time invariant regulatorapproaching the desired operating condition along the optimal path; andtransitioning control of the air-to-ground guided munition to themission autopilot.
 2. The method of claim 1, wherein control of theair-to-ground guided munition is transitioned to the mission autopilotafter a specific interval of time.
 3. The method of claim 2, wherein thespecific interval of time is one second.
 4. A method of guiding anair-to-ground guided munition, between release from a platform and anactivation of a mission autopilot along an optimal path, the methodcomprising: determining a nominal reference trajectory that optimizes adesired performance index for the air-to-ground guided munition usingoptimal control theory by: determining an optimal trajectory includingan optimal flight path and open-loop inputs using a high fidelity model;and generating a local model by linearizing the high fidelity modelalong a prescribed flight path, wherein the local model is used todetermine time-varying feedback gains; implementing a feedback controlsystem that optimizes an original performance index to second order in apresence of disturbances along an optimal path using neighboring optimalcontrol; converging, via the feedback control system, to a linear timeinvariant regulator approaching the desired operating condition alongthe optimal path; and transitioning control of the air-to-ground guidedmunition to the mission autopilot.
 5. The method of claim 4, wherein thehigh fidelity model includes dynamic system information and safety andacceptability criteria.
 6. The method of claim 4, wherein control of theair-to-ground guided munition is transitioned to the mission autopilotafter a specific interval of time.
 7. The method of claim 6, wherein thespecific interval of time is one second.